168 research outputs found
Nice labeling problem for event structures: a counterexample
In this note, we present a counterexample to a conjecture of Rozoy and
Thiagarajan from 1991 (called also the nice labeling problem) asserting that
any (coherent) event structure with finite degree admits a labeling with a
finite number of labels, or equivalently, that there exists a function such that an event structure with degree
admits a labeling with at most labels. Our counterexample is based on
the Burling's construction from 1965 of 3-dimensional box hypergraphs with
clique number 2 and arbitrarily large chromatic numbers and the bijection
between domains of event structures and median graphs established by
Barth\'elemy and Constantin in 1993
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
Recommended from our members
Problems and results on linear hypergraphs
In this thesis, we tackle several problems involving the study of 3-uniform, linear hypergraphs satisfying some additional structural constraint.
We begin with a problem of Hrushovski concerning Latin squares satisfying a partial associativity condition. From an Latin square one can define a binary operation , and is associative if and only if is a group multiplication table. Hrushovski asked whether, if is only associative a positive proportion of the time, must still in some sense be close to a group multiplication table. This problem manifests a well-studied combinatorial theme, in which a local structural constraint is relaxed (first to a `99' version and then to a `1' version) and the global consequences of the relaxed constraints are analysed. We show that the partial associativity condition is sufficient to deduce powerful global information, allowing us to find within a large subset with group-like structure. Since Latin squares can be regarded as 3-uniform, linear hypergraphs, and the partial associativity condition can be formulated in terms of the count of a particular subhypergraph, we are able to apply purely combinatorial methods to a problem that touches algebra, model theory and geometric group theory.
We then take this problem further. A condition due to Thomsen provides a combinatorial constraint which, if satisfied by the Latin square , proves that is in fact the multiplication table of an abelian group. It is then natural to ask whether a relaxed version of this result is also attainable, and by extending our methods we are able to prove a result of this flavour. Since the combinatorial obstructions to commutativity of are far more complex than those for associativity, topological complications arise that are not present in the earlier work.
We also study a problem of Loh concerning sequences of triples of integers from satisfying a certain `increasing' property. Loh studied the maximum length of such a sequence, improving a trivial upper bound of to using the triangle removal lemma and conjecturing that a natural construction of length is best possible. We provide the first power-type improvement to the upper bound, showing that there exists such that the length is bounded by . By viewing the triples as edges in a 3-uniform hypergraph, the increasing property shows that the hypergraph is linear and provides further restrictions in terms of forbidden subhypergraphs. By considering this formulation, we provide links to various important open problems including the Brown--Erd\H os--S\'os conjecture.
Finally, we present a collection of shorter results. In work connecting to the earlier chapters, we resolve the Brown--Erd\H os--S\'os conjecture in the context of hypergraphs with a group structure, and show moreover that subsets of group multiplication tables exhibit local density far beyond what can be hoped for in general. In work less closely connected to the main theme of the thesis, we also answer a question of Leader, Mili\'cevi\'c and Tan concerning partitions of boxes, consider a problem on projective cubes in , and resolve a conjecture concerning a diffusion process on graphs
On embeddings of CAT(0) cube complexes into products of trees
We prove that the contact graph of a 2-dimensional CAT(0) cube complex of maximum degree can be coloured with at most
colours, for a fixed constant . This implies
that (and the associated median graph) isometrically embeds in the
Cartesian product of at most trees, and that the event
structure whose domain is admits a nice labeling with
labels. On the other hand, we present an example of a
5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes
which cannot be embedded into a Cartesian product of a finite number of trees.
This answers in the negative a question raised independently by F. Haglund, G.
Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the
computation of the bounds in Theorem 1. Some figures repaire
Central limit theorem for exponentially quasi-local statistics of spin models on Cayley graphs
Central limit theorems for linear statistics of lattice random fields
(including spin models) are usually proven under suitable mixing conditions or
quasi-associativity. Many interesting examples of spin models do not satisfy
mixing conditions, and on the other hand, it does not seem easy to show central
limit theorem for local statistics via quasi-associativity. In this work, we
prove general central limit theorems for local statistics and exponentially
quasi-local statistics of spin models on discrete Cayley graphs with polynomial
growth. Further, we supplement these results by proving similar central limit
theorems for random fields on discrete Cayley graphs and taking values in a
countable space but under the stronger assumptions of {\alpha}-mixing (for
local statistics) and exponential {\alpha}-mixing (for exponentially
quasi-local statistics). All our central limit theorems assume a suitable
variance lower bound like many others in the literature. We illustrate our
general central limit theorem with specific examples of lattice spin models and
statistics arising in computational topology, statistical physics and random
networks. Examples of clustering spin models include quasi-associated spin
models with fast decaying covariances like the off-critical Ising model, level
sets of Gaussian random fields with fast decaying covariances like the massive
Gaussian free field and determinantal point processes with fast decaying
kernels. Examples of local statistics include intrinsic volumes, face counts,
component counts of random cubical complexes while exponentially quasi-local
statistics include nearest neighbour distances in spin models and Betti numbers
of sub-critical random cubical complexes.Comment: Minor changes incorporated based on suggestions by referee
The satisfiability threshold for random linear equations
Let be a random matrix over the finite field with
precisely non-zero entries per row and let be a random vector
chosen independently of . We identify the threshold up to which the
linear system has a solution with high probability and analyse the
geometry of the set of solutions. In the special case , known as the
random -XORSAT problem, the threshold was determined by [Dubois and Mandler
2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], and the proof
technique was subsequently extended to the cases [Falke and Goerdt
2012]. But the argument depends on technically demanding second moment
calculations that do not generalise to . Here we approach the problem from
the viewpoint of a decoding task, which leads to a transparent combinatorial
proof
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